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Bistability Dynamics in Structured Ecological Models

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Bistability Dynamics in Structured Ecological Models CHAPTER 3 Bistability Dynamics in Structured Ecological Models Jifa Jiang1 and Junping Shi2,3 1Department of Mathematics, Shanghai Normal University, Shanghai 200092, P.R.China [email protected] 2Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, USA [email protected] 3School of Mathematics, Harbin Normal University, Harbin, Heilongjiang 150080, P.R.China Abstract. Alternative stable states exist in many important ecosystems, and gradual change of the environment can lead to dramatic regime shift in these systems (Beisner et.al. (2003), May (1977), Klausmeier (1999), Rietkerk et.al. (2004), and Scheffer et.al. (2001)). Exam- ples have been observed in the desertification of Sahara region, shift in Caribbean coral reefs, and the shallow lake eutrophication (Carpenter et.al. (1999), Scheffer et.al. (2003), and Scheffer et.al. (2001)). It is well-known that a social-economical system is sustain- able if the life-support ecosystem is resilient (Holling (1973) and Folke et.al. (2004)). Here resilience is a measure of the magnitude of disturbances that can be absorbed before a system centered at one locally stable equilibrium flips to another. Mathematical mod- els have been established to explain the phenomena of bistability and hysteresis, which provide qualitative and quantitative information for ecosystem managements and policy making (Carpenter et.al. (1999) and Peters et.al. (2004)). However most of these models of catastrophic shifts are non-spatial ones. A theory for spatially extensive, heterogeneous ecosystems is needed for sustainable management and recovery strategies, which requires a good understanding of the relation between system feedback and spatial scales (Folke et.al. (2004), Walker et.al. (2004) and Rietkerk et.al. (2004)). In this chapter, we survey some recent results on structured evolutionary dynamics including reaction-diffusion equations and systems, and discuss their applications to structured ecological models which display bistability and hysteresis. In Section 1, we review several classical non-spatial models with 33 34 BISTABILITY DYNAMICS IN STRUCTURED ECOLOGICAL MODELS bistability; we discuss their counterpart reaction-diffusion models in Section 2, and espe- cially diffusion-induced bistability and hysteresis. In Section 3, we introduce some abstract results and concrete examples of threshold manifolds (separatrix) in the bistable dynamics. 3.1 Non-structured models The logistic model was first proposed by Belgian mathematician Pierre Verhulst (Ver- hulst (1838)): dP P = aP 1 , a,N> 0. (3.1) dt − N Here a is the maximum growth rate per capita, and N is the carrying capacity. A more general logistic growth type can be characterized by a declining growth rate per capita function. However it has been increasingly recognized by population ecolo- gists that the growth rate per capita may achieve its peak at a positive density, which is called an Allee effect (see Allee (1938), Dennis (1989) and Lewis and Kareiva (1993)). An Allee effect can be caused by shortage of mates (Hopf and Hopf (1985), Veit and Lewis (1996)), lack of effective pollination (Groom (1998)), predator satu- ration (de Roos et.al. (1998)), and cooperative behaviors (Wilson and Nisbet (1997)). If the growth rate per capita is negative when the population is small, we call such a growth pattern a strong Allee effect (see Fig.3.1-c); if f(u) is smaller than the max- imum but still positive for small u, we call it a weak Allee effect (see Fig.3.1-b). In Clark (1991), a strong Allee effect is called a critical depensation and a weak Allee effect is called a noncritical depensation. A population with a strong Allee effect is also called asocial by Philip (1957). Most people regard the strong Allee effect as the Allee effect, but population ecologists have started to realize that an Allee effect may be weak or strong (see Wang and Kot (2001), Wang, Kot and Neubert (2002)). Some possible growth rate per capita functions were also discussed in Con- way (1983,1984). A prototypical model with Allee effect is dP P P M = aP 1 − , a,N> 0. (3.2) dt − N · M | | If 0 <M<N, then theequation is of strong Allee effecttype, and if N<M< 0, thenit is of weakAllee effecttype.At least in thestrongAllee effect case,− M is called the sparsity constant. The dynamics of the logistic equation is monostable with one globally asymptoti- cally stable equilibrium, and that of strong Allee effect is bistable with two stable equilibria. A weak Allee effect is also monostable, although the growth is slower at lower density. Another example of a weak Allee effect is the equation of higher order autocatalytic chemical reaction of Gray and Scott (1990): da db = kabp, = kabp, k> 0, p 1. (3.3) dt − dt ≥ Here a(t) and b(t) are the concentrations of the reactant A and the autocatalyst B, k is the reaction rate, and p 1 is the order of the reaction with respect to the ≥ NON-STRUCTURED MODELS 35 1.2 0.4 1 0.3 0.8 0.2 0.6 0.1 0.4 0.2 0.4 0.6 0.8 1 u 0.2 ±0.1 0 0.2 0.4 0.6 0.8 1 ±0.2 u ±0.2 0.15 0.1 0.2 0.05 0.1 0.2 0.4 0.6 0.8 1 u ±0.05 0 0.2 0.4 0.6 0.8 1 u ±0.1 ±0.1 0.08 0.1 u 0.06 0.2 0.4 0.6 0.8 1 0.04 0 0.02 ±0.1 0 0.2 0.4 0.6 0.8 1 ±0.2 ±0.02 u ±0.04 ±0.3 ±0.06 ±0.08 ±0.4 Figure 3.1 (a) logistic (top); (b) weak Allee effect (middle); (c) strong Allee effect (bottom); the graphs on the left are growth rate uf(u), and the ones on the right are growth rate per capita f(u). autocatalytic species. Notice that a(t)+ b(t) a0 + b0 is invariant, so that (3.3) can be reduced to ≡ db = k(a + b b)bp, k,a + b > 0, p 1, (3.4) dt 0 0 − 0 0 ≥ which is of weak Allee effect type if p > 1, and of logistic type if p = 1. An auto- catalytic chemical reaction has been suggested as a possible mechanism of various biological feedback controls (Murray (2003)), and the similarity between chemical reactions and ecological interactions has been observed since Lotka (1920) in his pioneer work. The cubic nonlinearity in (3.2) has also appeared in other biological models. One prominent example is the FitzHugh-Nagumo model of neural conduction (FitzHugh (1961) and Nagumo et.al. (1962)), which simplifies the classical Hodgkin-Huxley model: dv dw ǫ = v(v a)(1 v) w, = cv bw, ǫ,a,b,c> 0, (3.5) dt − − − dt − where v(t) is the excitability of the system (voltage), and w(t) is a recovery variable representing the force that tends to return the resting state. When c is zero and w =0, 36 BISTABILITY DYNAMICS IN STRUCTURED ECOLOGICAL MODELS (3.5) becomes (3.2). Another example is a model of the evolution of fecally-orally transmitted diseases by Capasso and Maddalena (1981/82, 1982): dz dz 1 = a z + a z , 2 = a z + g(z ), a ,a ,a > 0. (3.6) dt − 11 1 12 2 dt − 22 2 1 11 12 22 Here z1(t) denotes the (average) concentration of infectious agent in the environ- ment; z2(t) denotes the infective human population; 1/a11 is the mean lifetime of the agent in the environment; 1/a22 is the mean infectious period of the human in- fectives; a12 is the multiplicative factor of the infectious agent due to the human population; and g(z1) is the force of infection on the human population due to a concentration z1 of the infectious agent. If g(z1) is a monotone increasing concave function, then it is known that the system is monostable with the global asymptoti- cal limit being either an extinction steady state or a nontrivial endemic steady state. However if g(z1) is a monotone sigmoid function, i.e. a monotone convex-concave function with S-shape and saturating to a finite limit, then the system (3.6) possesses two nontrivial endemic steady states and the dynamics of (3.6) is bistable, which can be easily seen from the phase plane analysis. 1 0.8 0.6 V 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 r Figure 3.2 Equilibrium bifurcation diagram of (3.8) with h = 0.1, where the horizontal axis is r and the vertical axis is V . Now we turn to some existing models which could lead to catastrophic shifts in ecosystems. In 1960-70s, theoretical predator-prey systems are proposed to demon- strate various stability properties in systems of populations at two or more trophic levels (Rosenzweig and MacArthur (1963) and Rosenzweig (1971)). A simplified model with such a predator-preyfeature is that of a grazing system of herbivore-plant interaction as in Noy-Meir (1975), see also May (1977). Here V (t) is the vegetation biomass, and its quantity changes following the differential equation: dV = G(V ) Hc(V ), (3.7) dt − where G(V ) is the growth rate of vegetationin absence of grazing, H is the herbivore NON-STRUCTURED MODELS 37 population density, and c(V ) is the per capita consumption rate of vegetation by the herbivore. If G(V ) is given by the familiar logistic equation, and c(V ) is the Holling type II (p = 1) or III (p > 1) functional response function (Holling (1959)), then (3.7) has the form (after nondimensionalization): dV rV p = V (1 V ) , h,r> 0, p 1.
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